We prove that algebraic K-theory satisfies `pro-descent' for abstract blow-up squares of noetherian schemes. As an application we derive Weibel's conjecture on the vanishing of negative K-groups.Comment: 48 pages, final versio

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We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents differential algebraic K-theory classes by geometric vector bundles. As an application we derive Lott's relation between short exact sequences of geometric bundles with a higher analytic torsion form.

We introduce a variant of homotopy K-theory for Tate rings, which we call analytic K-theory. It is homotopy invariant with respect to the analytic affine line viewed as an ind-object of closed disks of increasing radii. Under a certain regularity assumption we prove an analytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms of models. Along the way we also prove some results about the algebraic K-theory of Tate rings.